As Yoshida put it: "Every time you take an action in the game, the game will communicate with the server, check for cheating and come back afterwards. This creates a delay that an offline game doesn't have, since there's no need to check on the Internet."
FFXIV has some designs to mitigate this. One is called "snapshots". If you're in the telegraph (orange circle) zone when the enemy's cast bar stops casting, you are "snapshotted" to that zone. Then the boss animation plays, sometimes a few seconds afterwards. So it doesn't matter what the boss does, if you were in the AOE, you're hit even if you're not visibly "hit" by a boss animation.
This design works when telegraphs exist, a cast bar exists, and there is ample time to move. If there are no telegraphs or cast bars, such as in the Fall Guys event, people expect to be hit by the animation instead, leading to a few seconds' miscalculation per above. Or, as in the new raids: there are telegraphs but not a cast bar. If mechanics are too fast, you have to move in and out of AOEs quickly without time to deliberate the "snapshot" period per the cast bar, thus the Fall Guys effect.
Furthermore, there's an even worse issue when considering the lag between your movement and the server recognizing it. You move a second before the server recognizes it. This means that, even if you dodge the "snapshot", you can still be caught in the snapshot, called "clipping". When mechanics are fast, like the new raids, this can happen a lot more often. It's usually found with high ping, but even Japanese players are having an issue with the new content because of faster and overlapping mechanics.
Any and all design issues with the new content can be fixed with better netcode. The challenge would be seen as fair (improvable with practice) rather than unfair (random and not responsive to skill) with zero change to the fight design itself. Otherwise, changes to design would have to be made to reduce the complexity of mechanics. As a certain other poster said, we have had easy content for "two expansions" (actually longer) because of this. For Dawntrail, the dev team decided to remove their old precautions. The above problems are a result.